Launch the Platform
To begin a time series analysis, choose the Time Series command from the Analyze > Specialized Modeling submenu to display the Time Series Launch dialog (Figure 15.1). This dialog allows you to specify the number of lags to use in computing the autocorrelations and partial autocorrelations. It also lets you specify the number of future periods to forecast using each model fitted to the data. After you select analysis variables and click OK on this dialog, a platform launches with plots and accompanying text reports for each of the time series (Y) variables you specified.
Select Columns into Roles
You assign columns for analysis with the dialog in Figure 15.1. The selector list at the left of the dialog shows all columns in the current table. To cast a column into a role, select one or more columns in the column selector list and click a role button. Or, drag variables from the column selector list to one of the following role boxes:
X, Time ID
for the x-axis, one variable used for labeling the time axis
Y, Time Series
for the y-axis, one or more time series variables.
If you use a X, Time ID variable, you can specify the time frequency by using the Time Frequency column property. The choices are Annual, Monthly, Weekly, Daily, Hourly, Minute, and Second. This helps JMP determine the spacing of the data when plotting the forecast values. If no frequency is specified, the data is treated as equally spaced numeric data.
To remove an unwanted variable from an assigned role, select it in the role box and click Remove. After assigning roles, click OK to see the analysis for each time series variable versus the time ID.
You set the number of lags for the autocorrelation and partial autocorrelation plots in the Autocorrelation Lags box. This is the maximum number of periods between points used in the computation of the correlations. It must be more than one but less than the number of rows. A commonly used rule of thumb for the maximum number of lags is n/4, where n is the number of observations. The Forecast Periods box allows you to set the number of periods into the future that the fitted models are forecast. By default, JMP uses 25 lags and 25 forecast periods.
The data for the next examples are in the Seriesg.jmp table found in the Time Series sample data folder (Box and Jenkins 1976). The time series variable is Passengers and the Time ID is Time.
Figure 15.1 The Time Series Launch Dialog
The Time Series Graph
The Time Series platform begins with a plot of each times series by the time ID, or row number if no time ID is specified (Figure 15.2). The plot, like others in JMP, has features to resize the graph, highlight points with the cursor or brush tool, and label points. See the JMP Reports chapter in the Using JMP book for a discussion of these features.
Figure 15.2 Time Series Plot of Seriesg (Airline Passenger) Data
If you open Time Series Basic Diagnostic Tables, graphs of the autocorrelation and partial autocorrelation (Figure 15.3) of the time series are shown.
The platform popup menu, discussed next, also has fitting commands and options for displaying additional graphs and statistical tables.
Time Series Commands
The platform red triangle menu has the options described in the following sections.
Graph
The Time Series platform begins by showing a time series plot, like the one shown previously in Figure 15.2. The Graph command on the platform popup menu has a submenu of controls for the time series plot with the following commands.
Time Series Graph
hides or displays the time series graph.
Show Points
hides or displays the points in the time series graph.
Connecting Lines
hides or displays the lines connecting the points in the time series graph.
Mean Line
hides or displays a horizontal line in the time series graph that depicts the mean of the time series.
Autocorrelation and Partial Autocorrelation
The autocorrelation graph describes the correlation between all the pairs of points in the time series for a given separation in time (lag). Autocorrelation and partial autocorrelation graphs can help you determine whether the time series is stationary (meaning it has a fixed mean and standard deviation over time) and what model might be appropriate to fit the time series.
Tip: The autocorrelation graph of the sample is often called the sample autocorrelation function.
Figure 15.3 Autocorrelation (left) and Partial Correlation (right) Graphs
Note the following about these graphs:
• Blue lines represent two standard errors.
• For autocorrelation, the Ljung-Box Q and p-values appear for each lag. The Q-statistic can be used to test whether a group of autocorrelations is significantly different from zero, or to test that the residuals from a model can be distinguished from white noise.
• The number of lags begins with 0 to provide a broader picture of the analysis. To compute correlations beginning with lag 1, modify the JMP preferences before generating the graph. Select File > Preferences > Platforms > Time Series and then select Suppress Lag 0 in ACF and PACF.
Statistical Details for Autocorrelation and Partial Autocorrelation
The autocorrelation for the kth lag is computed as follows:
where 
where 
is the mean of the N non-missing points in the time series. The bars graphically depict the autocorrelations.
is the mean of the N non-missing points in the time series. The bars graphically depict the autocorrelations.By definition, the first autocorrelation (lag 0) always has length 1. The curves show twice the large-lag standard error (± 2 standard errors), computed as
For partial autocorrelation, the blue lines represent ± 2 standard errors for approximate 95% prediction limits, where the standard error is computed as follows:
for all kVariogram
The Variogram command alternately displays or hides the graph of the variogram. The variogram measures the variance of the differences of points k lags apart and compares it to that for points one lag apart. The variogram is computed from the autocorrelations as
where rk is the autocorrelation at lag k. The plot on the left in Figure 15.4 shows the Variogram graph for the Seriesg data.
AR Coefficients
The AR Coefficients command alternately displays or hides the graph of the least squares estimates of the autoregressive (AR) coefficients. The definition of these coefficients is given below. These coefficients approximate those that you would obtain from fitting a high-order, purely autoregressive model. The right-hand graph in Figure 15.4 shows the AR coefficients for the Seriesg data.
Figure 15.4 Variogram Graph (left) and AR Coefficient Graph (right)
Spectral Density
The Spectral Density command alternately displays or hides the graphs of the spectral density as a function of period and frequency (Figure 15.5).
The least squares estimates of the coefficients of the Fourier series
and
where 
are combined to form the periodogram 
, which represents the intensity at frequency fi.
are combined to form the periodogram , which represents the intensity at frequency fi.The periodogram is smoothed and scaled by 1/(4π) to form the spectral density.
The Fisher’s Kappa statistic tests the null hypothesis that the values in the series are drawn from a normal distribution with variance 1 against the alternative hypothesis that the series has some periodic component. Kappa is the ratio of the maximum value of the periodogram, I(fi), and its average value. The probability of observing a larger Kappa if the null hypothesis is true is given by
where q = N / 2 if N is even, q = (N - 1) / 2 if N is odd, and κ is the observed value of Kappa. The null hypothesis is rejected if this probability is less than the significance level α.
For q - 1 > 100, Bartlett’s Kolmogorov-Smirnov compares the normalized cumulative periodogram to the cumulative distribution function of the uniform distribution on the interval (0, 1). The test statistic equals the maximum absolute difference of the cumulative periodogram and the uniform CDF. If it exceeds 
, then reject the hypothesis that the series comes from a normal distribution. The values a = 1.36 and a = 1.63 correspond to significance levels 5% and 1% respectively.
, then reject the hypothesis that the series comes from a normal distribution. The values a = 1.36 and a = 1.63 correspond to significance levels 5% and 1% respectively.Figure 15.5 Spectral Density Plots
Save Spectral Density
Save Spectral Density creates a new table containing the spectral density and periodogram where the (i+1)th row corresponds to the frequency fi = i / N (that is, the ith harmonic of 1 / N).
The new data table has these columns:
Period
is the period of the ith harmonic, 1 / fi.
Frequency
is the frequency of the harmonic, fi.
Angular Frequency
is the angular frequency of the harmonic, 2πfi.
Sine
is the Fourier sine coefficients, ai.
Cosine
is the Fourier cosine coefficients, bi.
Periodogram
is the periodogram, I(fi).
Spectral Density
is the spectral density, a smoothed version of the periodogram.
Difference
Many time series do not exhibit a fixed mean. Such nonstationary series are not suitable for description by some time series models such as those with only autoregressive and moving average terms (ARMA models). However, these series can often be made stationary by differencing the values in the series. The differenced series is given by the following equation:
where t is the time index and B is the backshift operator defined by Byt = yt-1.
The Difference command computes the differenced series and produces graphs of the autocorrelations and partial autocorrelations of the differenced series. These graphs can be used to determine whether the differenced series is stationary.
Several of the time series models described in the next sections accommodate a differencing operation (the ARIMA, Seasonal ARIMA models, and some of the smoothing models). The Difference command is useful for determining the order of differencing that should be specified in these models.
The Differencing Specification dialog appears in the report window when you select the Difference command. It enables you to specify the differencing operation that you want to apply to the time series. Click Estimate to see the results of the differencing operation. The Specify Differencing dialog enables you to specify the Nonseasonal Differencing Order, d, the Seasonal Differencing Order, D, and the number of Observations per Period, s. Selecting zero for the value of the differencing order is equivalent to no differencing of that kind.
The red triangle menu on the Difference plot has the following options:
Graph
controls the plot of the differenced series and behaves the same as those under the Time Series Graph menu.
Autocorrelation
alternately displays or hides the autocorrelation of the differenced series.
Partial Autocorrelation
alternately hides or displays the partial autocorrelations of differenced series.
Variogram
alternately hides or displays the variogram of the differenced series.
Save
appends the differenced series to the original data table. The leading d + sD elements are lost in the differencing process. They are represented as missing values in the saved series.
Decomposition
Decomposition of time series data isolates the linear trend and seasonal cycles, allowing for better model estimation. To model the underlying variation in time series data, you must first detrend and deseasonalize your data. For example, an increase in sales from October to December might simply be a result of seasonal variation in sales, not a meaningful increase.
Typically, you would begin decomposition by removing any linear trend, and then removing long cycles, such as a 12-month cycle. Then you could start removing short cycles, such as 6-month cycles.
The Decomposition option provides three ways of removing trend and cyclical effects from your data.
Remove Linear Trend
Removes the linear trend from the data, where Trend = β0 + β1 time. A Time Series report for the detrended series is added to the report window. Also, a column containing the most recently detrended series is added to the data table.
Note: When you select the Remove Linear Trend option, JMP adds a column to the data table that contains the detrended data. If this column is already present in the data table when the option is selected, JMP overwrites the existing column.
Remove Cycle
Removes cyclical effects from the data. You can specify the number of units per cycle and indicate whether a constant should be subtracted from the data. A Time Series report for the decycled series is added to the report window. Also, a column containing the most recently decycled series is added to the data table. See “Remove Cycle”.
Note: When you select the Remove Cycle option, JMP adds a column to the data table that contains the decycled data. If this column is already present in the data table when the option is selected, JMP overwrites the existing column.
X11
Removes trend and seasonal effects using the X-11 method developed by the US Bureau of the Census (Shiskin et. al., 1967). Available only for monthly or quarterly data. You can specify a multiplicative or additive X-11 adjustment. See “X11”.
Remove Cycle
This option estimates the cyclic component of a time series using a single cosine wave and then removes it from the data. The Define Cycle window appears when you select Remove Cycle. Enter the number of units per cycle and indicate whether a constant (or intercept) should be removed from the data. The cyclical component that is fit to the data has the following form:
The notation used is defined as follows:
• C is the (optional) Constant.
• A is the Amplitude of the cosine wave.
• U is the number of Units per Cycle.
• P is the Phase of the cosine wave.
• t is one less than the row number of a given observation.
The model is fit to the data. The predicted values are subtracted from the original time series.
A report is added showing the following:
• estimates for Amplitude, Units per Cycle, Phase, and Constant
• the equation for the cyclic component
• a Time Series Plot of the decycled series
• a Time Series Basic Diagnostics report for the decycled series.
A column containing the decycled data is added to the data table.
X11
Required Data Structure
The X11 option is available only for monthly or quarterly data. The X, Time ID column must contain numeric values equally spaced by month or quarter without any gaps or missing values. JMP returns an error if you request X11 for a time column that does not satisfy these requirements.
When regarded as number of seconds since January 1, 1904, the data in the X, Time ID column must represent exact monthly or quarterly increments. This is best accomplished by using a Time ID column that has a Date Format of m/y or yyyyQq. (See The Column Info Window chapter in the Using JMP book for information about Date Formats.)
Description
This method adjusts the original time series using either a multiplicative or an additive decomposition. The model is fit using an iterative process to estimate the three X-11 components: trend cycle, seasonal, and irregular. The trend cycle component contains both the long-term trend and the long-term cyclical effects. The irregular component contains the effects of variation unexplained by the trend and seasonal components. For a historical overview of the development of the X-11 method, see SAS/ETS 13.1 User’s Guide (search for “Historical Development of X-11”).
The multiplicative adjustment fits the following model:
where:
Ot is the original time series
Ct is the trend cycle component
St is the seasonal component
It is the irregular component
The adjusted multiplicative trend is 
.
.The additive adjustment fits the following model:
The adjusted additive trend is 
.
.X11 Reports
Depending on your selection of Decomposition Type, the X11 option adds an X11 - Multiplicative report or an X11 - Additive report. The reports consist of four plots:
Original and Adjusted
Overlays the X11-adjusted time series on the original time series, Ot. Figure 15.6 shows an example of the original and adjusted time series plot. The X11-adjusted values are 
for the multiplicative adjustment and 
for the additive adjustment.
for the multiplicative adjustment and for the additive adjustment.D10 - Final Seasonal Factors
Plots the seasonal factor components, St, over time.
D12 - Final Trend Cycle
Plots the trend cycle components, Ct, over time.
D13 - Final Irregular Series
Plots the irregular components, It, over time.
Figure 15.6 Original and Adjusted Plot for X11 Analysis for Monthly Sales.jmp
X11 Report Options
The X11 reports have the following red triangle options:
Show Tables
Shows or hides the X11 summary tables, as described in Shiskin et. al. (1967). The tables are grouped into five categories (labeled B through F), described in Table 15.1.
Save Columns
Saves four columns to the data table: the seasonally adjusted time series, the seasonal components (St), the trend cycle components (Ct), and the irregular series components (It).
Save All Columns
Saves columns to the data table for all of the tables produced in the report by the Show Tables option.
|
Letter Prefix
|
Category Description
|
|
B
|
preliminary estimates of seasonal, trend cycle, and irregular components
|
|
C
|
intermediate estimates of seasonal, trend cycle, and irregular components
|
|
D
|
final estimates of seasonal, trend cycle, and irregular components
|
|
E
|
analytical tables
|
|
F
|
summary measures
|
For details about the contents of the X11 output tables, see Shisken et. al. (1967) or SAS/ETS 13.1 User’s Guide (search for “The output from PROC X11”).
Show Lag Plot
The plot show how an observation at time t is related to another observation at time t +/- p. +/- p is the lag, because the plot allows negative values and positive values.
Number of Forecast Periods
The Number of Forecast Periods command displays a dialog for you to reset the number of periods into the future that the fitted models will forecast. The initial value is set in the Time Series Launch dialog. All existing and future forecast results will show the new number of periods with this command.
Modeling Reports
The time series modeling commands are used to fit theoretical models to the series and use the fitted model to predict (forecast) future values of the series. These commands also produce statistics and residuals that allow you to ascertain the adequacy of the model you have elected to use. You can select the modeling commands repeatedly. Each time you select a model, a report of the results of the fit and a forecast is added to the platform results.
The fit of each model begins with a dialog that lets you specify the details of the model being fit as well as how it will be fit. Each general class of models has its own dialog, as discussed in their respective sections. The models are fit by maximizing the likelihood function, using a Kalman filter to compute the likelihood function. The ARIMA, seasonal ARIMA, and smoothing models begin with the following report tables.
Model Comparison Table
Figure 15.7 shows the Model Comparison Report.
Figure 15.7 Model Comparison
The Model Comparison table summarizes the fit statistics for each model. You can use it to compare several models fitted to the same time series. Each row corresponds to a different model. The models are sorted by the AIC statistic. The Model Comparison table shown above summarizes the Seasonal ARIMA (0, 1, 1)(0, 1, 1)12 and ARIMA models (1, 0, 0), (0, 0, 1), (1, 0, 1), and (1, 1, 1) respectively. Use the Report checkbox to show or hide the Model Report for a model.
The Weights column shows the normalized AIC weight. An AIC weight is calculated as follows:
BestAIC is the smallest AIC value among the fitted models. The AIC weights are then normalized by dividing the sum of all AIC weights of all models in the comparison table. The AIC Weights are then sorted in decreasing order.
Model Comparison Options
In the Model Comparison report, each model has the following red triangle menu options:
Fit New
opens a window giving the settings of the model. You can change the settings to fit a different model.
Simulate Once
provides one simulation of the model out k time periods. The simulation is shown on the Model Comparison time series plot. To change k, use the Number of Forecast Periods option on the platform red-triangle menu.
Simulate More
provides the specified number of simulations of the model out k time periods. The simulations are shown on the Model Comparison time series plot. To change k, use the Number of Forecast Periods option on the platform red-triangle menu.
Remove Model Simulation
removes the simulations for the given model.
Remove All Simulation
removes the simulations for all models.
Generate Simulation
generates simulations for the given model, and stores the results in a data table. You specify the random seed, number of simulations, and the number of forecast periods.
Set Seed
is used to specify the seed for generating the next forecasts.
The Model Comparison report provides plots for a model when the Graph checkbox is selected. Figure 15.8 shows the plots for the Seasonal ARIMA (0, 1, 1)(0, 1, 1)12 model.
Figure 15.8 Model Plots
The top plot is a time series plot of the data, forecasts, and prediction limits. Below that are plots of the autocorrelation and partial autocorrelation functions.
Model Summary Table
Each model fit generates a Model Summary table, which summarizes the statistics of the fit. In the formulae below, n is the length of the series and k is the number of fitted parameters in the model.
DF
is the number of degrees of freedom in the fit, n – k.
Sum of Squared Errors
is the sum of the squared residuals.
Variance Estimate
the unconditional sum of squares (SSE) divided by the number of degrees of freedom, SSE / (n – k). This is the sample estimate of the variance of the random shocks at, described in the section “ARIMA Model”.
Standard Deviation
is the square root of the variance estimate. This is a sample estimate of the standard deviation of at, the random shocks.
Akaike’s Information Criterion [AIC], Schwarz’s Bayesian Criterion [SBC or BIC]
Smaller values of these criteria indicate better fit. They are computed:
RSquare
RSquare is computed
where 
and 
, 
are the one-step-ahead forecasts, and 
is the mean yi.
and , are the one-step-ahead forecasts, and is the mean yi.If the model fits the series badly, the model error sum of squares, SSE might be larger than the total sum of squares, SST and R2 will be negative.
RSquare Adj
The adjusted R2 is
MAPE
is the Mean Absolute Percentage Error, and is computed
MAE
is the Mean Absolute Error, and is computed
–2LogLikelihood
Twice the negative log-likelihood function evaluated at the best-fit parameter estimates. Smaller values are better fits. See the Statistical Details appendix in the Fitting Linear Models book.
Stable
indicates whether the autoregressive operator is stable. That is, whether all the roots of 
lie outside the unit circle.
lie outside the unit circle.Invertible
indicates whether the moving average operator is invertible. That is, whether all the roots of 
lie outside the unit circle.
lie outside the unit circle.Parameter Estimates Table
There is a Parameter Estimates table for each selected fit, which gives the estimates for the time series model parameters. Each type of model has its own set of parameters. They are described in the sections on specific time series models. The Parameter Estimates table has these terms:
Term
lists the name of the parameter. These are described below for each model type. Some models contain an intercept or mean term. In those models, the related constant estimate is also shown. The definition of the constant estimate is given under the description of ARIMA models.
Factor (Seasonal ARIMA only)
lists the factor of the model that contains the parameter. This is only shown for multiplicative models. In the multiplicative seasonal models, Factor 1 is nonseasonal and Factor 2 is seasonal.
Lag (ARIMA and Seasonal ARIMA only)
lists the degree of the lag or backshift operator that is applied to the term to which the parameter is multiplied.
Estimate
lists the parameter estimates of the time series model.
Std Error
lists the estimates of the standard errors of the parameter estimates. They are used in constructing tests and prediction intervals.
t Ratio
lists the test statistics for the hypotheses that each parameter is zero. It is the ratio of the parameter estimate to its standard error. If the hypothesis is true, then this statistic has an approximate Student’s t-distribution. Looking for a t-ratio greater than 2 in absolute value is a common rule of thumb for judging significance because it approximates the 0.05 significance level.
Prob>|t|
lists the observed significance probability calculated from each t-ratio. It is the probability of getting, by chance alone, a t-ratio greater (in absolute value) than the computed value, given a true hypothesis. Often, a value below 0.05 (or sometimes 0.01) is interpreted as evidence that the parameter is significantly different from zero.
The Parameter Estimates table also contains the Constant Estimate, for models that contain an intercept or mean term. The definition of the constant estimate is given under “ARIMA Model”.
If an ARIMA or seasonal ARIMA model specifies an intercept term, the Parameter Estimates table shows the intercept value in a column labeled Mu.
Forecast Plot
Each model has its own Forecast plot. The Forecast plot shows the values that the model predicts for the time series. It is divided by a vertical line into two regions. To the left of the separating line the one-step-ahead forecasts are shown overlaid with the input data points. To the right of the line are the future values forecast by the model and the prediction intervals for the forecasts.
You can control the number of forecast values by changing the setting of the Forecast Periods box in the platform launch dialog or by selecting Number of Forecast Periods from the Time Series drop-down menu. The data and prediction intervals can be toggled on and off using the Show Points and Show Prediction Interval commands on the model’s popup menu.
Residuals
The graphs under the residuals section of the output show the values of the residuals based on the fitted model. These are the actual values minus the one-step-ahead predicted values. In addition, the autocorrelation and partial autocorrelation of these residuals are shown. These can be used to determine whether the fitted model is adequate to describe the data. If it is, the points in the residual plot should be normally distributed about the zero line and the autocorrelation and partial autocorrelation of the residuals should not have any significant components for lags greater than zero.
Iteration History
The model parameter estimation is an iterative procedure by which the log-likelihood is maximized by adjusting the estimates of the parameters. The iteration history for each model you request shows the value of the objective function for each iteration. This can be useful for diagnosing problems with the fitting procedure. Attempting to fit a model which is poorly suited to the data can result in a large number of iterations that fail to converge on an optimum value for the likelihood. The Iteration History table shows the following quantities:
Iter
lists the iteration number.
Iteration History
lists the objective function value for each step.
Step
lists the type of iteration step.
Obj-Criterion
lists the norm of the gradient of the objective function.
Model Report Options
The title bar for each model you request has a popup menu, with the following options for that model:
Show Points
hides or shows the data points in the forecast graph.
Show Prediction Interval
hides or shows the prediction intervals in the forecast graph.
Save Columns
creates a new data table with columns representing the results of the model.
Save Prediction Formula
saves the data and prediction formula to a new data table.
Create SAS Job
creates SAS code that duplicates the model analysis in SAS.
Submit to SAS
submits code to SAS that duplicates the model analysis. If you are not connected to a SAS server, prompts guide you through the connection process.
Residual Statistics
controls which displays of residual statistics are shown for the model. These displays are described in the section “Time Series Commands”; however, they are applied to the residual series.
ARIMA Model
An AutoRegressive Integrated Moving Average (ARIMA) model predicts future values of a time series by a linear combination of its past values and a series of errors (also known as random shocks or innovations). The ARIMA command performs a maximum likelihood fit of the specified ARIMA model to the time series.
For a response series 
, the general form for the ARIMA model is:
, the general form for the ARIMA model is:
where
t is the time index
B is the backshift operator defined as
is the response series after differencingμ is the intercept or mean term.
and 
, respectively, the autoregressive operator and the moving average operator and are written
and
are the sequence of random shocks.The 
are assumed to be independent and normally distributed with mean zero and constant variance.
are assumed to be independent and normally distributed with mean zero and constant variance.The model can be rewritten as
where the constant estimate δ is given by the relation 
.The ARIMA command displays the Specify ARIMA Model dialog, which allows you to specify the ARIMA model you want to fit. The results appear when you click Estimate.
Use the Specify ARIMA Model dialog for the following three orders that can be specified for an ARIMA model:
1. The Autoregressive Order is the order (p) of the polynomial 
operator.
operator.2. The Differencing Order is the order (d) of the differencing operator.
3. The Moving Average Order is the order (q) of the differencing operator 
.
.4. An ARIMA model is commonly denoted ARIMA(p,d,q). If any of p,d, or q are zero, the corresponding letters are often dropped. For example, if p and d are zero, then model would be denoted MA(q).
The Prediction Intervals box allows you to set the prediction level between 0 and 1 for the forecast prediction intervals. The Intercept check box determines whether the intercept term μ will be part of the model. If the Constrain fit check box is checked, the fitting procedure will constrain the autoregressive parameters to always remain within the stable region and the moving average parameters within the invertible region. You might want to uncheck this box if the fitter is having difficulty finding the true optimum or if you want to speed up the fit. You can check the Model Summary table to see if the resulting fitted model is stable and invertible.
Seasonal ARIMA
In the case of Seasonal ARIMA modeling, the differencing, autoregressive, and moving average operators are the product of seasonal and nonseasonal polynomials:
where s is the number of observations per period. The first index on the coefficients is the factor number (1 indicates nonseasonal, 2 indicates seasonal) and the second is the lag of the term.
The Seasonal ARIMA dialog appears when you select the Seasonal ARIMA command. It has the same elements as the ARIMA dialog and adds elements for specifying the seasonal autoregressive order (P), seasonal differencing order (D), and seasonal moving average order (Q). Also, the Observations per Period box lets you specify the number of observations per period (s). The seasonal ARIMA models are denoted as Seasonal ARIMA(p,d,q)(P,D,Q)s.
ARIMA Model Group
The ARIMA Model Group option on the platform red-triangle menu allows the user to fit a range of ARIMA or Seasonal ARIMA models by specifying the range of orders. Figure 15.9 shows the dialog.
Figure 15.9 ARIMA Group
Transfer Functions
This example analyzes the gas furnace data (seriesJ.jmp) from Box and Jenkins. To begin the analysis, select Input Gas Rate as the Input List and Output CO2 as the Y, Time Series. The launch dialog should appear as in Figure 15.10.
Figure 15.10 Series J Launch Dialog
When you click OK, the report in Figure 15.11 appears.
Report and Menu Structure
This preliminary report shows diagnostic information and groups the analysis in two main parts. The first part, under Time Series Output CO2, contains analyses of the output series, while the Input Time Series Panel, contains analyses on the input series. The latter may include more than one series.
Figure 15.11 Series J Report
Each report section has its own set of commands. For the output (top) series, the commands are accessible from the red triangle on the outermost outline bar (Transfer Function Analysis). For the input (bottom) series, the red triangle is located on the inner outline bar (Input Series: Input Gas Rate).
Figure 15.12 shows these two command sets. Note their organization. Both start with a Graph command. The next set of commands are for exploration. The third set is for model building. The fourth set includes functions that control the platform.
Figure 15.12 Output and Input Series Menus
Diagnostics
Both parts give basic diagnostics, including the sample mean (Mean), sample standard deviation (Std), and series length (N).
In addition, the platform tests for stationarity using Augmented Dickey-Fuller (ADF) tests.
Zero Mean ADF
tests against a random walk with zero mean, i.e.
Single Mean ADF
tests against a random walk with a non-zero mean, i.e.
Trend ADF
tests against a random walk with a non-zero mean and a linear trend, i.e.
Basic diagnostics also include the autocorrelation and partial autocorrelation functions, as well as the Ljung-Box Q-statistic and p-values, found under the Time Series Basic Diagnostics outline node.
The Cross Correlation command adds a cross-correlation plot to the report. The length of the plot is twice that of an autocorrelation plot, or 
.
.Figure 15.13 Cross Correlation Plot
The plot includes plots of the output series versus all input series, in both numerical and graphical forms. The blue lines indicate two standard errors.
Model Building
Building a transfer function model is quite similar to building an ARIMA model, in that it is an iterative process of exploring, fitting, and comparing.
Before building a model and during the data exploration process, it is sometimes useful to prewhiten the data. This means find an adequate model for the input series, apply the model to the output, and get residuals from both series. Compute cross-correlations from residual series and identify the proper orders for the transfer function polynomials.
To prewhiten the input series, select the Prewhitening command. This brings up a dialog similar to the ARIMA dialog where you specify a stochastic model for the input series. For our SeriesJ example, we use an ARMA(2,2) prewhitening model, as shown in Figure 15.14.
Figure 15.14 Prewhitening Dialog
Click Estimate to reveal the Prewhitening plot.
Patterns in these plots suggest terms in the transfer function model.
Transfer Function Model
A typical transfer function model with m inputs can be represented as
where
Yt denotes the output series
X1 to Xm denote m input series
et represents the noise series
X1, t–d1 indicates the series X1 is indexed by t with a d1-step lag
μ represents the mean level of the model
ϕ(B) and θ(B) represent autoregressive and moving average polynomials from an ARIMA model
ωk(B) and δk(B) represent numerator and denominator factors (or polynomials) for individual transfer functions, with k representing an index for the 1 to m individual inputs.
Each polynomial in the above model can contain two parts, either nonseasonal, seasonal, or a product of the two as in seasonal ARIMA. When specifying a model, leave the default 0 for any part that you do not want in the model.
Select Transfer Function to bring up the model specification dialog.
Figure 15.15 Transfer Function Specification Dialog
The dialog consists of several parts.
Noise Series Orders
contains specifications for the noise series. Lowercase letters are coefficients for non-seasonal polynomials, and uppercase letters for seasonal ones.
Choose Inputs
lets you select the input series for the model.
Input Series Orders
specifies polynomials related to the input series.The first three orders deal with non-seasonal polynomials. The next four are for seasonal polynomials. The final is for an input lag.
In addition, there are three options that control model fitting.
Intercept
specifies whether μ is zero or not.
Alternative Parameterization
specifies whether the general regression coefficient is factored out of the numerator polynomials.
Constrain Fit
toggles constraining of the AR and MA coefficients.
Forecast Periods
specifies the number of forecasting periods for forecasting. If there are rows at the end of the data table that contain missing values for the Y variable and nonmissing values for the input variables, these rows are used in the initial forecasting settings. The values for the input variables are treated as future values of the input variables.
Using the information from prewhitening, we specify the model as shown in Figure 15.15.
Model Reports
The analysis report is titled Transfer Function Model and is indexed sequentially. Results for the Series J example are shown in Figure 15.16.
Figure 15.16 Series J Transfer Function Reports
Model Summary
gathers information that is useful for comparing models.
Parameter Estimates
shows the parameter estimates and is similar to the ARIMA version. In addition, the Variable column shows the correspondence between series names and parameters. The table is followed by the formula of the model. Note the notation B is for the backshift operator.
Residuals, Iteration History
are the same as their ARIMA counterparts.
Interactive Forecasting
provides a forecasting graph based on a specified prediction interval.
A prediction interval around the prediction is shown in blue. The confidence level for this prediction interval can be changed by entering a number in the Prediction Interval box above the graph.
You can drag the plus sign in the graph to change the number of forecast periods in the graph.
You can change the values of the input values in the following ways:
‒ Use the Import Inputs from Table button to change the values of the input variables.
‒ Drag points in the input graph to different values.
The results of your changes are reflected in real time in the Interactive Forecasting graph.
The following commands are available from the report drop-down menu.
Save Columns
creates a new data table containing the input and output series, a time column, predicted output with standard errors, residuals, and 95% prediction limits. This table contains missing values for the input and output columns for the forecast periods.
Create SAS Job
creates PROC ARIMA code that can reproduce this model.
Submit to SAS
submits PROC ARIMA code to SAS that reproduces the model.
Model Comparison Table
The model comparison table works like its ARIMA counterpart by accumulating statistics on the models you specify.
Fitting Notes
A regression model with serially correlated errors can be specified by including regressors in the model and not specifying any polynomial orders.
Intervention analysis can also be conducted, but prewhitening is no longer meaningful.
Currently, the transfer function model platform has limited capability of supporting missing values.
Smoothing Models
JMP offers a variety of smoothing techniques.
Smoothing models represent the evolution of a time series by the model:
where
is the time-varying mean term,
is the time-varying slope term,
is one of the s time-varying seasonal terms,
are the random shocks.Models without a trend have 
and nonseasonal models have 
. The estimators for these time-varying terms are
and nonseasonal models have . The estimators for these time-varying terms are
smoothed level that estimates
is a smoothed trend that estimates
for 
are the estimates of the 
.Each smoothing model defines a set of recursive smoothing equations that describes the evolution of these estimators. The smoothing equations are written in terms of model parameters called smoothing weights. They are
α, the level smoothing weight
γ, the trend smoothing weight
ϕ, the trend damping weight
δ, the seasonal smoothing weight.
While these parameters enter each model in a different way (or not at all), they have the common property that larger weights give more influence to recent data while smaller weights give less influence to recent data.
Each smoothing model has an ARIMA model equivalent. You may not be able to specify the equivalent ARIMA model using the ARIMA command because some smoothing models intrinsically constrain the ARIMA model parameters in ways the ARIMA command will not allow.
Simple Moving Average
A simple moving average model (SMA) produces plotted values that are equal to the average of consecutive observations in a time window. The plotted values can be uncentered or centered in the time window.
If we let ft=(yt+yt-1+yt-2+...yt-(k-2)+yt-(k-1))/k be the average of k consecutive observations, then:
• for an uncentered SMA(k), ft is the plotted value for time t.
• for a centered SMA(k) with odd k, ft is the plotted value for time t-(k-1)/2.
• for a centered SMA(k) with even k, ft is the plotted value for time t-(k-1)/2. When saved to a data table, ft is at t-(k-2)/2.
• for a centered SMA(k) with double smoothing for even k, the smoothing estimates are given by the following:
To fit a simple moving average model, select Smoothing > Simple Moving Average. A window appears with the following options:
Enter smoothing window width
Enter the width of the smoothing window.
Centering
Choose between No Centering, Centered, and Centered and Double Smoothed for Even Number of Terms.
The Simple Moving Average report shows a time plot of the data and the fitted model. The red triangle menu has the following options:
Add Model
Select this option to fit another model. When additional models are fit, the model is added to the time plot of the data.
Save to Data Table
Saves the original data, and plotted values of all moving average models.
Show Points
Shows or hides the points on the plot.
Connecting Lines
Shows or hides the lines on the plot.
Smoothing Model Dialog
The Smoothing Model dialog appears in the report window when you select one of the smoothing model commands.
The Prediction Intervals popup list allows you to set the prediction level for the forecast prediction intervals. The dialogs for seasonal smoothing models include a Observations per Period box for setting the number of observations in a period. The Constraints popup list lets you to specify what type of constraint you want to enforce on the smoothing weights during the fit. The constraints are:
Zero To One
keeps the values of the smoothing weights in the range zero to one.
Unconstrained
allows the parameters to range freely.
Stable Invertible
constrains the parameters such that the equivalent ARIMA model is stable and invertible.
Custom
expands the dialog to allow you to set constraints on individual smoothing weights. Each smoothing weight can be Bounded, Fixed, or Unconstrained as determined by the setting of the popup menu next to the weight’s name. When entering values for fixed or bounded weights, the values can be positive or negative real numbers.
The example shown here has the Level weight (α) fixed at a value of 0.3 and the Trend weight (γ) bounded by 0.1 and 0.8. In this case, the value of the Trend weight is allowed to move within the range 0.1 to 0.8 while the Level weight is held at 0.3. Note that you can specify all the smoothing weights in advance by using these custom constraints. In that case, none of the weights would be estimated from the data although forecasts and residuals would still be computed. When you click Estimate, the results of the fit appear in place of the dialog.
Simple Exponential Smoothing
The model for simple exponential smoothing is 
.
.The smoothing equation, Lt = αyt + (1 – α)Lt-1, is defined in terms of a single smoothing weight α. This model is equivalent to an ARIMA(0, 1, 1) model where
with 
.The moving average form of the model is
Double (Brown) Exponential Smoothing
The model for double exponential smoothing is 
.
.The smoothing equations, defined in terms of a single smoothing weight α are
and 
.This model is equivalent to an ARIMA(0, 1, 1)(0, 1, 1)1 model
where 
with 
.The moving average form of the model is
Linear (Holt) Exponential Smoothing
The model for linear exponential smoothing is 
.
.The smoothing equations defined in terms of smoothing weights α and γ are
and
This model is equivalent to an ARIMA(0, 2, 2) model where
with 
and 
.The moving average form of the model is
Damped-Trend Linear Exponential Smoothing
The model for damped-trend linear exponential smoothing is 
.
.The smoothing equations in terms of smoothing weights α, γ, and ϕ are
and
This model is equivalent to an ARIMA(1, 1, 2) model where
with 
and 
.The moving average form of the model is
Seasonal Exponential Smoothing
The model for seasonal exponential smoothing is 
.
.The smoothing equations in terms of smoothing weights α and δ are
and
This model is equivalent to a seasonal ARIMA(0, 1, 1)(0, 1, 0)S model where we define
, 
, and
so
with
, 
, and 
.The moving average form of the model is
where
Winters Method (Additive)
The model for the additive version of the Winters method is 
.
.The smoothing equations in terms of weights α, γ, and δ are
, 
, and 
.This model is equivalent to a seasonal ARIMA(0, 1, s+1)(0, 1, 0)s model
The moving average form of the model is
where
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